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Mathematical Analysis of Graph and Huffman Coding Problems

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Dr Anjan Krishnamurthy
Dr Anjan Krishnamurthy

This document summarizes and analyzes algorithms for minimum spanning tree problems and Huffman coding. It discusses Prim's and Kruskal's algorithms for finding minimum spanning trees in graphs, and Huffman's algorithm for data compression. For each algorithm, it covers the key steps, proofs of correctness, and mathematical analysis of time complexity. It also provides context on problems addressed by minimum spanning trees and motivation for using Huffman coding in data compression.

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Mathematical Analysis of Graph and Huffman Problems Anjan.K II Sem M.Tech CSE M.S.R.I.T 06/10/09 DAA: Analysis of Graph and Huffman Problem
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem
Key Points – Recap ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem
Graph Problems-MST ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem
Prim-Jarnik’s MST ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem
Prim-Jarnik’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem
Prim-Jarnik’s – Proof of Correctness 06/10/09 DAA: Analysis of Graph and Huffman Problem Theorem: Upon termination of Algorithm , T is a MST Proof is by induction. Given a spanning tree T if an unique edge f is not in the tree, adding which the tree form a unique cycle and e some edge like f then T = T U {f} – {e}. ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Prim-Jarnik’s Algorithm Analysis ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem
06/10/09 DAA: Analysis of Graph and Huffman Problem
Different Ways To Implement Priority Queue 06/10/09 DAA: Analysis of Graph and Huffman Problem
Kruskal’s MST ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem
Kruskal’s – Proof of Correctness 06/10/09 DAA: Analysis of Graph and Huffman Problem Theorem: Upon termination of Algorithm , forest F is a MST Proof is by induction. Given a set of nodes S if an unique edge f is not in the tree, adding which the forest form a unique cycle and e some edge like f then F = S U {f} – {e}. ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Kruskal’s Algorithm Analysis ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem
06/10/09 DAA: Analysis of Graph and Huffman Problem
Data Compression using Huffman Coding ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem
Information Theory - Entropy ,[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem
Huffman Algorithm 06/10/09 DAA: Analysis of Graph and Huffman Problem
Huffman Coding – Proof of Correctness ,[object Object],[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem
Proof (Cont’d) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem
Huffman’s Algorithm Analysis ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem
06/10/09 DAA: Analysis of Graph and Huffman Problem
Summary ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem
References ,[object Object],[object Object],[object Object],[object Object],06/10/09 DAA: Analysis of Graph and Huffman Problem

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Mathematical Analysis of Graph and Huffman Coding Problems

Notes de l'éditeur

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